M.M.o.t.C.P
Michael Eung-Min Lee
St. John's College
University of Oxford
 |
| Mathematical Models of The Carding Proces |
1 Introduction 1
1.1 Textile Manufacture . . . . 1
1.1.1 The Crosrol Revolving-Flats Carding Machine . . 3
1.1.1.1 From Feeder-in to Taker-in . . 5
1.1.1.2 On the Taker-in . . . .. 5
1.1.1.3 From Taker-in to Cylinder . . . . 6
1.1.1.4 On the Cylinder . . . . . . . 7
1.1.1.5 The Do er . . . . . 8
1.1.1.6 Summary . . . . . . . . 8
1.1.2 Literature review . . . . .. . . 8
1.2 Thesis Overview . . . . . . .. . . . . 14
2 Aerodynamics and Single Fibres 16
2.1 Introduction . . . . . . . 16
2.2 A Fibre in the Carding Machine . . . . 17
2.3 A Mathematical Model for a Single Fibre . 18
2.3.1 Drag on a Fibre . . 18
2.3.2 Internal Fibre Forces . .. . . 20
2.3.3 Aerodynamics . . . . 22
2.3.4 Important Parameters . . . 23
2.4 Fibres on the Cylinder and Taker-in . . . 24
2.4.1 Rotational Forces . . . . . . 25
2.4.2 Fluid Dynamics . . . .. 26
2.4.2.1 Annular Flow without Hooks . . . 27
2.4.2.2 Annular Flow with Hooks . . . . 28
2.4.3 The Equations . . . . . . 30
2.4.3.1 Boundary Conditions . . . . 32
2.4.4 Asymptotic Solution . . . . . 33
2.4.4.1 Annular Flow without Hooks . . .33
2.4.4.2 Annular Flow with Hooks .. . . 33
2.4.5 Numerical Computations . . .. . 34
2.4.5.1 Annular Flow without Hooks . . 34
2.4.5.2 Annular Flow with Hooks . . 36
2.4.5.3 Friction between and bre and a hook . 39
2.4.6 The Do er . . . . . 40
2.4.7 Summary . . . . . 41
2.5 Transfer Mechanisms .. . . . 42
2.5.1 Aerodynamics .. . . 42
2.5.1.1 From Taker-In to Cylinder (Strong Transfer) . 45
2.5.1.2 From Cylinder to Do er (Weak Transfer) . . 46
2.5.2 Motion of a bre at a transfer point. . 47
2.5.2.1 Solutions . . . . 49
2.5.3 Frictional Contact Points . . . 51
2.6 Conclusion . . . . . 52
3 Tufts and Fibres 54
3.1 Introduction . . . 54
3.2 The Fibres . . . . . . . . 56
3.3 The Withdrawal of a Single Fibre . . . 57
3.3.1 Constitutive Law . . . . 60
3.3.2 Friction . . . . . 61
3.3.3 The Equations . . . . . . .. 61
3.3.3.1 Dimensionless Equations . . 63
3.3.4 Asymptotic Solutions . . . . 64
3.3.4.1 Small Asymptotics .. . . 65
3.3.4.2 Small Time Solution for = O(1) . . 67
3.3.5 Numerical Computations . . . . 69
3.4 Teasing out Fibres with a Hook . . . . 73
3.5 Tufts held together by a single bre . .. . 75
3.5.1 The Equations . . . . . 78
3.5.1.1 Dimensionless Equations . . . 80
3.5.2 Asymptotic Solutions . . . . 81
3.5.2.1 Small Asymptotics .. . . 81
3.6 Tuft breaking . . . . . . 85
3.7 Conclusion . . . . 87
4 Continuum Models for Interacting Fibres 89
4.1 Introduction .. . 90
4.2 Experiments . . . . 92
4.3 A Simple Viscous Model . . . 95
4.3.1 Problem Formulation . . . 95
4.3.1.1 Fibre Contact Points .. . 96
4.3.2 Dimensionless Equations . 98
4.3.3 The Extensional Simulation . . . . 99
4.4 A Continuum Model with Direction and Alignment .. 102
4.4.1 The Governing Equations .. . . 104
4.4.1.1 The Stress Tensor . . . 105
4.4.2 Kinematic Condition . . . 112
4.4.3 Empirical Law for the Order Parameter . . 113
4.4.4 The Two Dimensional Equations . . . 114
4.4.5 Elongation of a Fibrous Mass. . . 115
4.4.5.1 The Governing Equations . . . 115
4.4.5.2 Boundary Conditions . . 116
4.4.5.3 Dimensionless Lagrangian Formulation . . 117
4.4.5.4 The Solution for a Uniformly Dense Tuft. . 118
4.5 Continuum Model with Entanglement . . . . . 121
4.5.1 Degree of Entanglement and Braid Theory . . . . 122
4.5.2 Governing Equations. . 125
4.5.2.1 Empirical Law for Entanglement .126
4.5.2.2 The Stress Tensor . . 126
4.5.3 Elongation of a Fibrous Mass. . 127
4.5.3.1 The Governing Equations . 127
4.5.3.2 Boundary Conditions . 127
4.5.3.3 Dimensionless Lagrangian Formulation. . 128
4.5.3.4 The Solution For a Uniformly Dense Tuft . 129
4.5.3.5 Comparison with Experiment . . 130
4.5.4 A Simple Shearing Problem . 133
4.5.4.1 The Governing Equations . 133
4.5.4.2 Dimensionless Formulation . 134
4.5.4.3 The Solution . .. 135
4.5.4.4 Comparison with Experiments . . 138
4.5.5 An Array of Hooks . .. . . . . .138
4.6 Conclusion . 139
5 Conclusions 141
5.1 The Life of Fibres in the Carding Machine 142
5.1.1 The Taker-In . . 143
5.1.2 The Cylinder. 143
5.1.3 The Do er .144
5.1.4 Suggested Further Work. . 144
A Dimensional and Dimensionless Numbers 146A.1 Drum Speeds . 146
A.2 The Fibres. 146
A.3 Fluid Dynamics and Drag . 146
A.3.1 Stokes Drag . 147
A.3.2 Taylor Drag. 148
B Shear Breaking Experiments on Tufts 149
C Stability Analysis of a Fibre in the Carding Machine 153
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A few years ago, a group from the School of Textile Industries in Leeds approached the
Oxford Centre for Industrial and Applied Mathematics with the long-standing problem
of understanding bre dynamics in carding. Although the machines involved have
undergone numerous modi cations, predominantly fuelled by the advances in mechanical
manufacturing technology, the rudiments behind the process have not changed for
centuries. Furthermore, research and development in this \low-technology" industry
have hitherto depended on empirical evidence. Within a multi-disciplinary group,
including industrialists and experimentalists, we have endeavoured to shed light on
this age-old and fundamental process.
This thesis presents the theoretical aspects
.........................................
This thesis presents the theoretical aspects of the work accomplished within this
multi-disciplinary research framework. Bespoke mathematical models have been derived
for the bres on three di erent length scales. The rst of the scales models the
motion of a single bre as it travels through the carding machine. The intermediate
scale focuses on the interplay between bres and a tuft in order to understand how
bres are extracted or teased out of tufts. Finally, we consider large volumes of interacting
bres as a continuum, and consider their evolution throughout the process.
These models as a whole give us access to a theoretical simulation that caters for all
areas of the machine at least to a rst approximation. We describe the process in
more detail and review the theoretical work done so far and then give a more detailed
overview of the thesis.
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Product details
Price
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File Size
| 3,189 KB |
Pages
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174 p |
File Type
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PDF format |
Title
| Mathematical Models of The Carding Proces |
Author
| Michael Eung-Min Lee |
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